We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Kor\'{ a}nyi unit ball $B$ in the first Heisenberg group $\mathbb{ H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Kor\'{ a}nyi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{ H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential maximal function of $f$. As an application, we characterize Carleson measures on $B$ via integral inequalities for quasiconformal mappings on $B$ and their radial limits. Our paper thus extends results by Astala and Koskela, Jerison and Weitsman, Nolder, and Zinsmeister, from $\mathbb{ R}^n$ to $\mathbb{ H}^1$. A crucial difference between the proofs in $\mathbb{ R}^n$ and $\mathbb{ H}^1$ is caused by the nonisotropic nature of the Kor\'{ a}nyi unit sphere with its two characteristic points.
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